Matrix polynomial

Polynomial means many terms. Now that we are able to multiply a matrix by a scalar and find powers of matr ices, we can fomi matrix polynomial equations, for example. 

Note that the matrix polynomial in A can be factored to give (A-51) and (A-tl)... 

The general form for a matrix polynomial equation satisfied by A is... 

To evaluate the matrix polynomial in Eq. (9-23), we use the MATLAB function polyvalm () which applies the coefficients in p2 to the matrix A. 

The recursion (84) can be extended to operators and matrices. This is done by using the Cayley-Hamilton theorem [2], which states that for a given analytic scalar function/(m), the expression for its operator counterpart/(U) is obtained via replacement of u by U as in Eq. (6). In this way, we can introduce the Lanczos operator and matrix polynomials defined by the following recursions ... 

However, it is just this situation which is expressed by the term tautomerism " of valencies. Depending upon the influence of the static graph on the matrix polynomial, one is able to distinguish between isomerism and tautomerism or bond fluctuation and mesomerism, respectively. 

Let P[[ ]] be a matrix polynomial of arbitrary degree that we wish to compute. Then P(A) represents the corresponding polynomial of A. A theorem of algebra states that there exist polynomials (A) and r(A) such that... 

We also define the matrix polynomial product, using the symbol o as the operator ... 

We will use the matrix polynomial product in the context of DWT factorisation (see Chapter 7). 

G. H. Golub, The use of Chebyshev matrix polynomials in the iterative solution of linear equations compared to the method of sujccessive overrelaxation. Doctoral Thesis, University of Illinois, 1959. 

Knot invariants are used in knot theory in order to characterize, distinguish, and classify topological properties of knots. A knot invariant is a function of a knot which takes the same value for all equivalent knots. There are numerical, matrix, polynomial, and finite-type invariants. In this section, the apphcation of some numerical and polynomial invariants to textiles will be... 

Since A is a matrix polynomial of the source term (A = - AsourceY + (Ajource) ) and the linearized reaction term is singular because there is a single reaction term, the eigenvalues of A can be computed according to (Gantmacher 1959) ... 

The two eigenstates of light emerging from a homo- As shown by Schonhofer and co-workers A can be re-geneous sample superpose in accordance with their written as a matrix polynomial of third degree different optical path lengths. The azimuth and the el-lipticity are then given by the Stokes vector... 

In the work of King, Dupuis, and Rys [15,16], the mabix elements of the Coulomb interaction term in Gaussian basis set were evaluated by solving the differential equations satisfied by these matrix elements. Thus, the Coulomb matrix elements are expressed in the form of the Rys polynomials. The potential problem of this method is that to obtain the mabix elements of the higher derivatives of Coulomb interactions, we need to solve more complicated differential equations numerically. Great effort has to be taken to ensure that the differential equation solver can solve such differential equations stably, and to... 

A square matrix has the eigenvalue A if there is a vector x fulfilling the equation Ax = Ax. The result of this equation is that indefinite numbers of vectors could be multiplied with any constants. Anyway, to calculate the eigenvalues and the eigenvectors of a matrix, the characteristic polynomial can be used. Therefore (A - AE)x = 0 characterizes the determinant (A - AE) with the identity matrix E (i.e., the X matrix). Solutions can be obtained when this determinant is set to zero. 

A tircial solution to this equation is x = 0. For a non-trivial solution, we require that the deterniinant A - AI equals zero. One way to determine the eigenvalues and their associated eigenvectors is thus to expend the determinant to give a polynomial equation in A. Ko." our 3x3 symmetric matrix this gives ... 

The described direct derivation of shape functions by the formulation and solution of algebraic equations in terms of nodal coordinates and nodal degrees of freedom is tedious and becomes impractical for higher-order elements. Furthermore, the existence of a solution for these equations (i.e. existence of an inverse for the coefficients matrix in them) is only guaranteed if the elemental interpolations are based on complete polynomials. Important families of useful finite elements do not provide interpolation models that correspond to complete polynomial expansions. Therefore, in practice, indirect methods are employed to derive the shape functions associated with the elements that belong to these families. 

The degree of the least polynomial of a square matr ix A, and henee its rank, is the number of linearly independent rows in A. A linearly independent row of A is a row that eannot be obtained from any other row in A by multiplieation by a number. If matrix A has, as its elements, the eoeffieients of a set of simultaneous nonhomo-geneous equations, the rank k is the number of independent equations. If A = , there are the same number of independent equations as unknowns A has an inverse and a unique solution set exists. If k < n, the number of independent equations is less than the number of unknowns A does not have an inverse and no unique solution set exists. The matrix A is square, henee k > n is not possible. 

The method of finding uncertainty limits for linear equations can be generalized to higher-order polynomials. The matrix method for finding the minimization... 

Polynomial root finding, as in the previous section, has some technical pitfalls that one would like to avoid. It is easier to write reliable software for matrix diagonalization (QMOBAS, TMOBAS) than it is for polynomial root finding hence, diagonalization is the method of choice for Huckel calculations. 

If, in particular, we convert a matrix L —xl into its SCF, where now the (0,1)-entries of L are elements of -Fig], and factor the similarity invariants into products of powers of monic irreducible polynomials Pj x), so that fi x) =... 

Recalling that the companion matrix of any monic polynomial, e x)... 

Before this is done, however, a certain paradox needs to be discussed briefly. Given a matrix A, and a nonsingular matrix V, it is known that A, and V XA V, have the same characteristic polynomial, and the two matrices are said to be similar. Among all matrices similar to a given matrix A, there are matrices of the form... 

The methods of simple and of inverse iteration apply to arbitrary matrices, but many steps may be required to obtain sufficiently good convergence. It is, therefore, desirable to replace A, if possible, by a matrix that is similar (having the same roots) but having as many zeros as are reasonably obtainable in order that each step of the iteration require as few computations as possible. At the extreme, the characteristic polynomial itself could be obtained, but this is not necessarily advisable. The nature of the disadvantage can perhaps be made understandable from the following observation in the case of a full matrix, having no null elements, the n roots are functions of the n2 elements. They are also functions of the n coefficients of the characteristic equation, and cannot be expressed as functions of a smaller number of variables. It is to be expected, therefore, that they... 

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